The probability density function (PDF) of the solution process of a nonlinear stochastic differential equation (SDE) is found in this paper using the path integration technique. The SDE is a piecewise linear system representing a model of an imperfectly mounted spur gear pair with a small stochastic noise added to the driving force. It is known that the system model for a particular choice of parameters shows chaotic behavior (Kahraman and Singh, 1990, “Non-Linear Dynamics of a Spur Gear Pair,” J. Sound Vibrat., 142(1), pp. 49–75). The PDF is compared with the Poincaré map of the deterministic system and it is shown that the stochastic and deterministic attractors are very similar. Then it is shown that although the stochastic attractor appears clearly after just a few iterations, the probability density over the attractor depends on the initial condition. The system does converge to one unique periodic PDF eventually but the convergence is fairly slow. However, the transient is almost periodic with a period that is twice that of the forcing, which can be utilized to obtain a much higher convergence rate. The advantage of using a SDE to study this rattling problem is that it can provide a very detailed picture of the dynamics and the most likely states of the system can immediately be identified.
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July 2009
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Nonsmooth Dynamics by Path Integration: An Example of Stochastic and Chaotic Response of a Meshing Gear Pair
E. Mo,
E. Mo
Department of Mathematical Sciences,
Norwegian University of Science and Technology
, Trondheim NO-7491, Norway
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A. Naess
A. Naess
Department of Mathematical Sciences, Centre for Ships and Ocean Structures,
Norwegian University of Science and Technology
, Trondheim NO-7491, Norway
Search for other works by this author on:
E. Mo
Department of Mathematical Sciences,
Norwegian University of Science and Technology
, Trondheim NO-7491, Norway
A. Naess
Department of Mathematical Sciences, Centre for Ships and Ocean Structures,
Norwegian University of Science and Technology
, Trondheim NO-7491, NorwayJ. Comput. Nonlinear Dynam. Jul 2009, 4(3): 034501 (4 pages)
Published Online: May 20, 2009
Article history
Received:
August 23, 2007
Revised:
August 28, 2008
Published:
May 20, 2009
Citation
Mo, E., and Naess, A. (May 20, 2009). "Nonsmooth Dynamics by Path Integration: An Example of Stochastic and Chaotic Response of a Meshing Gear Pair." ASME. J. Comput. Nonlinear Dynam. July 2009; 4(3): 034501. https://doi.org/10.1115/1.3124780
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